Problem: What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
Explanation: Among five consecutive odd numbers, at least one is divisible by 3 and exactly one is divisible by 5, so the product is always divisible by 15. The cases $n=2$, $n=10$, and $n=12$ demonstrate that no larger common divisor is possible, since $\boxed{15}$ is the greatest common divisor of $3\cdot5\cdot7\cdot9\cdot11$, $11\cdot13\cdot15\cdot17\cdot19$, and $13\cdot15\cdot17\cdot19\cdot21$.